Method for Position Estimation Using Generalized Error Distributions

ABSTRACT

A method for improving the results of radio location systems that incorporate weighted least squares optimization generalizes the weighted least squares method by using maximum a posteriori (MAP) probability metrics to incorporate characteristics of the specific positioning problem (e.g., UTDOA). Weighted least squares methods are typically used by TDOA and related location systems including TDOA/AOA and TDOA/GPS hybrid systems. The incorporated characteristics include empirical information about TDOA errors and the probability distribution of the mobile position relative to other network elements. A technique is provided for modeling the TDOA error distribution and the a priori mobile position. A method for computing a MAP decision metric is provided using the new probability distribution models. Testing with field data shows that this method yields significant improvement over existing weighted least squares methods.

CROSS-REFERENCE

This application is a continuation of U.S. patent application Ser. No.12/346,146, filed Dec. 30, 2008, currently pending.

TECHNICAL FIELD

The present application relates generally to the field of wirelesslocation, i.e., systems and methods for estimating the position of awireless device, and more particularly to a method using generalizederror distributions.

BACKGROUND

As the Federal Communications Commission (FCC) moves towards aPSAP-level location accuracy mandate, improving methods for differentlocation technologies becomes a necessity. The subject matter describedherein relates to the fields of communications and location technology.It provides a means for improving the accuracy of location technologiessuch as Global Positioning System (GPS), Uplink Time Difference ofArrival (UTDOA) and Advanced Forward Link Trilateration (AFLT).

A common approach to position estimation is to find a weighted leastsquares solution from measured quantities such as time differences,pseudoranges or power levels. The weighted least squares solution isknown to achieve a maximum likelihood (ML) solution when input errorsare independent and Gaussian (see J. Caffery, Wireless Location in CDMACellular Radio Systems, Boston-London: Kluwer Academic Publishers,2000), but it cannot do this under the more general conditionsencountered in practice. For example, TDOA errors have a tendency to bepositive relative to the predicted leading edge of the multipath delayprofile. As explained below, several factors such as imperfect leadingedge detection and non-line-of-sight (NLOS) propagation contribute tothese positive errors. As a result, the per-baseline error distributionis skewed. This skew reduces the accuracy of the basic weighted leastsquares method. In contrast, the method described herein exploitsknowledge of this skew to obtain improved results. Moreover, correlationamong these errors can often be found; for example, distinct multipathcomponents can be received at the same sector, common NLOS conditionsmay exist at a site and common errors may be introduced by the referencesignal. These correlations may be incorporated into a maximum aposteriori (MAP) algorithm as described below. This framework can alsobe used to incorporate an estimate of the a priori mobile positiondistribution in the location solution.

UTDOA is a network-based technology allowing for any signal transmittedfrom any type of mobile station (MS) to be received at any base stationto obtain a UTDOA measurement. A reference base station measures thereceived signal at roughly the same time as each cooperating basestation, as illustrated in FIG. 1.

FIG. 1 shows an idealized model of the signals available to or from amobile device for positioning where the present invention could be usedto increase the accuracy of the positioning estimate (also called alocation attempt). This figure also identifies the system components forwireless location. In FIG. 1, a Global Navigation Satellite System 101(GNSS) such as the United States NavStar Global Positioning System(GPS), broadcasts well defined, code division multiple access (CDMA)spread spectrum signals 107 used by specially equipped mobile wirelessdevices 102 for TDOA location estimation of latitude, longitude,altitude and velocity. If the mobile device 102 is not equipped toreceive satellite signals 107 for location calculation, both uplink anddownlink terrestrial wireless techniques using TDOA or Time-of-Arrival(TOA) calculations may be used to provide a location estimate.Terrestrial wide area wireless location techniques using downlink(network based transmitter-to-device) TDOA or TOA techniques includeAdvanced Forward Link Trilateration (FLT) [IS-95, IS-2000], EnhancedTime Difference of Arrival (E-OTD) [GSM] and Observed Time Difference ofArrival (OTDOA) [UMTS] as well as distributed beacon techniques.Terrestrial Downlink techniques require that the mobile device 102measure the downlink radio signals 108 from network based transmitters103 104 and then use the radio link(s) 109, backhaul facilities 113 andthe Wireless Communications Network 110 to convey the collected radiomeasurements to a Position Determining Entity (PDE) 106 for conversioninto a latitude, longitude and in some cases an altitude.

Terrestrial wide area wireless location techniques using uplink(device-to-network based receiver) TDOA or TOA techniques includeU-TDOA, U-TDOA/Angle of Arrival (AoA) hybrid and U-TDOA/Assisted GPS.U-TDOA and hybrids currently function in CDMA [IS-95, IS-2000], GSM,UMTS, WiMAX (802.16e/m and 802.20) and conceptually for the upcomingLong-Term-Evolution (LTE) OFDM based wireless radio access network(RAN). Terrestrial Uplink techniques require that the mobile device 102transmissions 109 be measured by network based receivers (in this caseco-located within the cell sites 103 104. Measurement data is thenconveyed by backhaul 111 to a Position Determining Entity (PDE) 106 forconversion into a latitude, longitude, velocity, and in some cases analtitude. Regardless of the aforementioned wireless location technique,determination of the radio signal time-of-flight is key to accuratedetermination of the mobile devices 102 actual location. In FIG. 1, thereal world influences of signal reflection, diffraction, and attenuationdue to constructive or destructive interference are not shown.

In the system of FIG. 1, by cross-correlating the received signal at thereference base station with the received signal at a cooperating basestation, a time difference of arrival is determined. The cooperatingstations send their TDOA measurements to a position determining entity(PDE) where a location solution is found. However, impairments to themeasurement can arise from additive noise and signal level fluctuations.These impairments may affect the sensitivity of detecting the presenceof the mobile signal at the cooperating base station. Other impairmentsto the estimation impact the cooperator's ability to detect the line ofsight (LOS) path delay.

FIGS. 2 a, 2 b, 2 c and 2 d illustrate how objects, such as a building,may block the direct path, creating a non-line of sight impairment indifferent location environments, including uplink, downlink, GNSS andhybrid GNSS/uplink systems (where GNSS stands for Global NavigationSatellite System). A diffracted path traveling around a building arrivesat the receiver later than the highly attenuated or completely blockeddirect path. Additionally, reflections from obstacles can causescattering, which produces dispersion of the arrival times of differentpaths. In FIG. 2 a, an example of an uplink wireless location system isdepicted. The mobile device 102 transmits a signal 109. In some cases,such as for the Reference Receiver 203, the radio signal is receiveddirectly (a line-of-sight or LOS case). But other receivers 104 mayreceive diffracted signal 202 or a reflected signal 203. In each casethe original uplink signal 109 may also be received or have the originalsignal blocked, attenuated or delayed by an obstruction 201.

FIG. 3 illustrates impairments that make detection of the first arrivaldifficult and cause a skewing of the TDOA error. The reference numeralsin FIG. 3 are used as follows:

-   303=Transmit time-   304=Detection threshold-   305=Line-of-sight (LOS) time-of-flight-   306=Lag time-   307=Basis for reported TOA or TDOA-   308=Delay spread-   309=Missed signal components

FIG. 3 shows the arrival times of a multi-path degraded signal on anamplitude 302 to time 301 plot 300. A signal is transmitted at time 303and has a potential direct path time-of-flight shown as 305. Theearliest signal component arrivals are undetected since they arrive at apower level below the detection threshold 304. The detection threshold304 must be maintained to avoid excessive false alarms. Missed earliestarrival detection events cause a reported TOA or TDOA that is largerthan the LOS TOA or TDOA which is desired. In this example the firstsignal above threshold 307 produces a lag of 306 from the true firstarriving signal component. Additionally, the earliest arriving multipathcomponents may arrive later than expected due to NLOS propagationcreating an NLOS delay. This also causes a reported TDOA that is largerthan the LOS TDOA. These factors skew errors between the TDOAmeasurements and an LOS TDOA being searched or computed by positioningalgorithms. Positioning decisions in the inventive solution describedherein exploit both the skewing of errors caused by these factors aswell as the non-Gaussian shape of the error distribution.

The method described in U.S. Pat. No. 6,564,065, May 13, 2003, K. Changet al., “Bayesian-update based location prediction method for CDMASystems,” appears to predict power levels from CDMA pilot channelmeasurements with location decisions made from an a posteriori powerdistribution using simulation. The method described in U.S. Pat. No.5,252,982, Oct. 12, 1993, E. Frei, “Method of precise positiondetermination,” appears to assume Gaussian errors using a weighted leastsquares method that iteratively finds phase ambiguities for a GPSlocation solution using an a posteriori RMS error.

SUMMARY

A method for improving the results of radio location systems thatincorporate weighted least squares optimization generalizes the weightedleast squares method by using maximum a posteriori (MAP) probabilitymetrics to incorporate characteristics of the specific positioningproblem (e.g., UTDOA). As discussed, WLS methods are typically used byTDOA and related location systems including TDOA/AOA and TDOA/GPS hybridsystems. The incorporated characteristics include empirical informationabout TDOA errors and the probability distribution of the mobileposition relative to other network elements. A technique is provided formodeling the TDOA error distribution and the a priori mobile position. Amethod for computing a MAP decision metric is provided using the newprobability distribution models.

An illustrative implementation provides an error detection methodcomprising: obtaining field data, wherein said field data have baselineor location dependent values to be used in said signal correlationmodel; analyzing said field data to obtain (1) a signal correlationmodel and associated measurement parameters, (2) correlation matrixrules, and (3) a model for a priori position; computing weights for themeasurements based on an estimated variability of the measurement; usingthe weights along with the correlation matrix rules to generate acovariance matrix, and computing an inverse covariance matrix;performing an iterative search over a geographical region to find alocation with a maximum a posteriori (MAP) metric; determining that astopping condition has been reached; and reporting the geographicposition with the largest MAP metric as the location solution.

The methods described herein include several key innovations, includingbut not necessarily limited to the following:

Analytical a priori distribution: Empirical data providing the actuallocations are used to obtain a distribution for the normalized distancefrom the reference tower to the location solution in order to model thegeneral shape of the a priori position relative to towers in the searcharea. An exponential distribution is shown to approximate the shape ofthe a priori position distribution and its variance is calculated fromthe empirical data.

Analytical TDOA error distribution: The double exponential distributionmodel is generalized to incorporate a skew and an arbitrary power in theexponent. Model parameters are estimated from empirical data.

Multipath/NLOS error indicators: Key indicators of the TDOA errordistribution include the number of baselines, the predicted multipathcorrection (based on observed signal parameters and/or knowledge of thelocal RF environment) and the TDOA correlation of each baseline. Methodsare provided to derive model parameters from these indicators byanalyzing empirical data and generating conditional error distributions.For each baseline, model parameters such as the skew are computed fromthese indicators.

TDOA error correlation: Methods are provided for computing a posterioriprobabilities for correlated errors between baselines that have theabove analytical TDOA error distribution. These correlations areincorporated into the MAP algorithm through the corresponding jointerror probability distribution.

Method for common bias mitigation: With more general distributions, itbecomes difficult to find an analytical solution for the common biasthat can exist in measurements. Methods for removal of the bias areprovided along with various complexity-performance tradeoffs.

Iterative adjustment: An iterative procedure is developed that appliesthe above methods. This procedure includes initialization operations andestimation of residual values.

Other features of the inventive technology are described below.

BRIEF DESCRIPTION OF THE DRAWINGS

The attached drawings include the following:

FIG. 1: Illustration of a positioning network.

FIGS. 2 a, 2 b, 2 c, and 2 d: Illustration of impairments to LOS pathdelay estimation.

FIG. 3: Illustration of causes for measurement error skewing.

FIGS. 4 a and 4 b: Components of MAP error detection method.

FIG. 5: Error distribution modeling process.

FIG. 6: Logic flow for the a priori distribution data analysis.

FIG. 7: Comparison of a priori location distribution with empiricalmodel.

FIG. 8: Conditional error distribution data analysis.

FIG. 9: Sample overall of error distribution vs. weighted least squaresmethod.

FIG. 10: Sample overall error distribution vs. new coarse model.

FIG. 11: Sample dependency of skew ratio to correlation.

FIG. 12: Sample overall conditional error distribution vs. new model(small skew).

FIG. 13: Sample conditional error distribution vs. new model (largeskew).

FIG. 14: Logic flow of MAP decision metric computation.

FIG. 15: Bias calculation for double exponential assumption (p=1,r=1).

FIG. 16: Conditional error calculation.

DETAILED DESCRIPTION OF ILLUSTRATIVE EMBODIMENTS

FIGS. 4 a-4 b show the components of an illustrative implementation ofthe MAP error detection method. As shown, the MAP process is started atstep 401. Field data 402 is analyzed 403 to obtain a set of signalcorrelation rules and models. These models and associated measurementparameters are developed from field data that can have baseline orlocation (or position, where the terms location and position are usedinterchangeably herein) dependent values to be used in the model. Forexample, the error skew may be higher for low correlation UTDOAmeasurements. A table 405 may thus be generated. This table provides amapping between the model parameter for the skew and the correlationvalue for the measurement. Similarly, a model and table is computed forthe a priori location 406. The field data analysis process also analyzesthe correlation between different receiver ports linking the locationreceiver (e.g., a Location Measuring Unit (LMU) or Signal CollectionSystem (SCS)) to an external antenna, providing correlation values andrules for their application. For example, there may be small correlationof errors on ports at the same location (co-site ports) due to NLOSeffects. Once the field data is analyzed, weights are computed for themeasurements based on an estimated variability of the measurement. Thenthe weights are used along with the correlation matrix rules to generatea port by port covariance matrix, which is inverted at 409.

As shown in FIG. 4 b, an iterative search over a geographical region isthen performed where the goal is to find the location with the largestMAP metric. Once the iterative search is initiated 411, a resolutionloop is entered 412 in which the geographic search space resolution isreduced in each iteration and new test points are generated viainterpolation. The search may be re-centered at the previous iteration'sminimum error point before proceeding. The current geographical regionis searched 414 by computing the MAP metric 415 for each test point inthe region and selecting the point with the smallest metric. The MAPmetric computation uses the covariance matrix 404, the error models andthe measurement parameter tables 405. If more test points exist in thesearch space, the processing logic 417 loops back to restart the search414. If no test points are untested in the present search space providedby the current resolution, the MAP process checks the pre-set resolutionlimit set 418. If the highest resolution has not been reached, theprocess returns to step 412; otherwise the MAP process checks if one ormore stopping criteria has/have been reached 419. When the stoppingcriteria are met, the MAP process ends 420 and the geographic positionwith the largest metric provides the location solution.

A goal of our inventive solutions is to model the a posterioriprobability of the error and find the location solution that maximizesthis probability. From the Bayes theorem (see A. Papoulis, ProbabilityRandom Variables, and Stochastic Processes, McGraw Hill Inc., New York,N.Y., 1984), the conditional probability density function of a randomposition vector, L, is given in terms of a vector of N measurementerrors, e, as

$\begin{matrix}{\mspace{79mu} {{{f_{L|e}\left( \underset{\_}{L} \middle| \underset{\_}{e} \right)} = \frac{{f_{e|L}\left( \underset{\_}{e} \middle| \underset{\_}{L} \right)}{f_{L}\left( \underset{\_}{L} \right)}}{f_{e}\left( \underset{\_}{e} \right)}}\mspace{20mu} {{where},}}} & (1) \\{\mspace{79mu} {\underset{\_}{L} \equiv {\begin{bmatrix}x \\y \\z\end{bmatrix}\mspace{14mu} {is}\mspace{14mu} a\mspace{20mu} {random}\mspace{14mu} {position}\mspace{14mu} {vector}}}} & (2) \\\begin{matrix}{\underset{\_}{e} = \begin{bmatrix}e_{1} \\e_{2} \\\ldots \\e_{N}\end{bmatrix}} \\{\equiv {\begin{bmatrix}{{\hat{\tau}}_{1} - {\tau_{1}\left( {x,y,z} \right)} - B} \\{{\hat{\tau}}_{2} - {\tau_{2}\left( {x,y,z} \right)} - B} \\\ldots \\{{\hat{\tau}}_{N} - {\tau_{N}\left( {x,y,z} \right)} - B}\end{bmatrix}\mspace{14mu} {is}\mspace{14mu} {the}\mspace{14mu} {TDOA}\mspace{14mu} {error}\mspace{14mu} {plus}\mspace{14mu} a\mspace{14mu} {common}\mspace{14mu} {bias}}}\end{matrix} & (3)\end{matrix}$

and

τ_(i)(x,y,z) is the LOS TDOA at point x,y,z for the ith baseline,

{circumflex over (τ)}_(i) is the ith TDOA baseline measurement, and

B is a common bias that may exist in the measurements

To simplify computations, the log of (1) can be maximized since theposition that maximizes (1) is also the position that maximizes the logof (1). The natural logarithm of (1) is

ln(f _(L|e)( L|e ))=ln(f _(e|L)( e|L ))+ln(f _(L)( L ))−ln(f _(e)( e))  (4)

Since the last term does not depend on the location it is constant whenconsidering different locations so it can be ignored. This leaves thefollowing function to be maximized over all locations:

ln(f _(L|e)( L|e ))=ln(f _(e|L)( e|L ))+ln(f _(L)( L ))  (5)

The first term is the log of the a posteriori error probability densityand the second term is the log of the a priori probability density.

Error Distribution Modeling Process

The error distribution modeling process is shown in FIG. 5. Initiallyfield data 402 is analyzed to determine the impact of various baselineand location specific measurement parameters on the a priori positiondistribution and the error distribution. Next, the correlation betweenerrors for different baselines is analyzed to determine correlationvalues and rules for obtaining these correlations. A coarse model forthe error distribution 504 is found that fits the overall errordistribution. A model for the a priori distribution is found based onthe a priori distribution of the field data. The error distributionmodel is then refined in order to be modifiable based on the variouslocation specific and baseline specific measurement parameters. Lastly,a correlation matrix is generated based on the correlation values andassociated rules for applying the values.

A Priori Distribution

The logic for finding an appropriate a priori distribution is shown inFIG. 6. Since location search areas can have vastly different dimensionsdepending on the location-specific data and location problems may alsohave vastly different network configurations, it is desirable to find adistribution of a model parameter that is a function of the actualposition in the search area. The position in the search area is the apriori location. It is desirable to find a model parameter that is afunction of the a priori location that can be computed and used todetermine an a priori location probability for use in equation (5).Initially, the field data is analyzed to determine candidate parameters.For example, the statistics for the distance from a reference tower maybe representative of the a priori location. Alternatively, the distancefrom a serving tower may be considered. These distances may benormalized to the maximum search range. For each parameter, the rangesand bin sizes must be chosen and histograms representing the a prioridistribution updated based on the actual (true) location in the fielddata. Each potential distribution is stored along with the parametercomputation in an a priori parameters database.

Once the potential a priori distributions are computed for various modelparameters, a model is selected as depicted in FIG. 5. The model ischosen to be both representative of the distribution from the field dataand computationally efficient. An exemplary model parameter is chosen tobe the distance of a candidate location from the reference towernormalized to the maximum distance from the reference tower to the edgeof the search region. This provides a transformation from the threedimensional random vector, L, to a normalized random distance from thereference tower, D, as

$\begin{matrix}{D = \frac{\left\lbrack {\left( {x - x_{ref}} \right)^{2} + \left( {y - y_{ref}} \right)^{2} + \left( {z - z_{ref}} \right)^{2}} \right\rbrack}{R_{\max}}} & (6)\end{matrix}$

where,

x_(ref),y_(ref),z_(ref) are the position coordinates of the referencetower

R_(max) is the maximum distance from the reference tower to the edge ofthe search region.

An exemplary model is chosen to be exponential as

$\begin{matrix}{{f_{D}(D)} = \left\{ \begin{matrix}{\lambda_{a}^{{- \lambda_{a}}D}} & {x \geq 0} \\0 & {Otherwise}\end{matrix} \right.} & (7)\end{matrix}$

where, λ_(a)=11 was chosen to fit the field data. FIG. 7 compares thismodel with the field data showing good agreement.

Error Distribution

The field data is also analyzed to obtain models for the errordistribution. FIG. 8 shows a series of steps similar to those in the apriori data analysis. In the figure, measurement parameters are analyzedto determine which ones cause large changes in the error distribution.Exemplary measurement parameters include:

UTDOA correlation for each baseline,

Multipath correction factor for each baseline,

Number of measurements for each location.

Ranges and bin sizes for these parameters are determined foraccumulating conditional and overall statistics. The conditionalstatistics and overall statistics are then compiled over all of thefield data and stored for model determination.

A sample overall of error distribution is shown in FIG. 9 and comparedwith the Gaussian distribution as would be assumed by weighted leastsquares methods. From the figure, it is evident that the Gaussianassumption does not adequately model the shape of the overalldistribution. A skewing of the overall distribution to the right isapparent.

The overall distribution provides input to the coarse error model asshown in FIG. 5. An exemplary error model for the ith marginal error isdetermined to be

$\begin{matrix}{{f_{e_{i}|L}\left( x_{i} \right)} = \left\{ \begin{matrix}{A\; ^{{- \frac{k}{\sigma_{i}^{p}}}{{r_{i}x_{i}}}^{p_{i}}}} & {x_{i} \geq 0} \\{A\; ^{{- \frac{k}{\sigma_{i}^{p}}}{\frac{x_{i}}{r_{i}}}^{p_{i}}}} & {x_{i} < 0}\end{matrix} \right.} & (8)\end{matrix}$

where,

p_(i) is a model parameter that is an arbitrary exponential powergreater than zero,

r_(i) is a model parameter that is a positive ratio indicating the skewof the distribution, and

σ_(i) is the standard deviation for the ith baseline.

∫_(−∞)^(∞)f_(e_(i)|L)(x) x = 1

Values for k and A are chosen that satisfy the condition for a givenr_(i) and p_(i). For a Gaussian distribution r_(i)=1, p_(i)=2 and k=½.For a double exponential distribution or Laplace distribution, r_(i)=1,p_(i)=1 and k=√{square root over (2)}.

The coarse modeling step computes values for the model parameters inequation (8) that match the field data. FIG. 10 illustrates the effectsof applying the coarse model with p_(i)=1.1 and r_(i)=1.1. The figureshows significantly better agreement with the field data than theGaussian assumption shown in FIG. 9.

The conditional error distributions are used as input to the “determinefine error model” block 506 in FIG. 5. For the conditional error, theskew ratio is computed based on the mean and standard deviation of theerror for each measurement parameter bin. The center of the bin is theconditional value for the error distribution.

The skew can be found in terms of the mean and standard deviation of theconditional distribution as follows. If the conditional errordistribution is approximated as a double exponential, then the scalingfactor in the exponent is

λ=√{square root over (2)}/σ  (9)

where, σ is the standard deviation of the conditional distribution.

To estimate r_(i), two separate scaled exponential distributions areconsidered where one of them is flipped around zero. Both components ofthe distribution are scaled to integrate to ½. As a result, the mean, m,of the conditional distribution can be put in terms of the scalingfactors as

$\begin{matrix}{m = {{- \frac{1}{\lambda_{L}}} + \frac{1}{\lambda_{R}}}} & (10)\end{matrix}$

where, λ_(L) and λ_(R) are the scaling factors in the exponents of theseparate exponential distribution components on the left and right ofzero respectively. It is assumed that all of the skew is due to changesin λ_(R) relative to λ allowing for the assumption λ_(L)≈λ. Solvingequation (10) for λ_(R) and using (9) with λ_(L)≈λ gives

$\begin{matrix}{\lambda_{R} = {\frac{1}{m + \frac{1}{\lambda_{L}}} \approx \frac{1}{m + \frac{\sigma}{\sqrt{2}}}}} & (11)\end{matrix}$

The skew ratio from (9) and (11) is then

$\begin{matrix}{r_{i} = {\frac{\lambda_{R}}{\lambda_{L}} = \frac{\sigma}{{\sqrt{2}m} + \sigma}}} & (12)\end{matrix}$

Values for σ and m from the conditional error distribution can then beused to compute r_(i) using equation (12).

An example of the skew ratio as a function of the UTDOA correlation isshown in FIG. 11. For each conditional distribution, the mean, m, andstandard deviation, σ, are used in equation (12) to compute r_(i). Inthe figure, it is evident that for low correlation, the skew increases.A linear fit to the data is shown which may be used in place of a tablelookup to average out the variability that is a function of the numberof samples and bin sizes. The result of the “fine error modeldetermination” block 506 (FIG. 5) is a mapping of measurement parametervalues, such as UTDOA correlation, to model parameters such as skew.Similar adjustments to the model parameter, p_(i), in (8) can be made asa function of the various measurement parameter values.

Exemplary results of the fine model adjustments are shown in FIG. 12where there is a small skew at higher correlation. FIG. 13 shows anexample when there is a large skew at lower correlation. These figuresillustrate that further improvement in the error distribution model isachieved relative to FIG. 10.

Correlation Matrix

The field data is also used to analyze the correlation that existsbetween errors for different ports as shown in FIG. 5. The correlationbetween errors is computed based on various rules or conditions.Exemplary rules for computing the correlation between two ports include:

Apply a fixed correlation between ports on the same sector,

Apply a fixed correlation between ports on the same site,

Apply a fixed correlation between a cooperating port and the referenceport.

For each rule a normalized correlation value or correlation coefficientfor the error is computed from the field data statistics (see A.Papoulis, Probability Random Variables, and Stochastic Processes, McGrawHill Inc., New York, N.Y., 1984). The correlation values and rulesprovide input to the “populate covariance matrix and perform inversion”block 409 (FIG. 4 a). The correlation matrix generation block computes aport by port matrix of correlation values. If multiple rules apply to apair of ports then the largest correlation value is used in the matrix.

Weighting and Variance Computation

A weighting for each baseline is based on the RMS error from the CramerRao bound (see R. McDonough, A. Whalen, Detection of Signals in Noise, 2nd Ed., Academic Press., San Diego, Calif., 1995). The lower bound onthe TDOA RMS error in AWGN (additive white Gaussian noise) is

$\begin{matrix}{\sigma_{i} \approx \frac{\sqrt{12}\rho_{i}}{2\pi \; {W\left( {2\; {WT}} \right)}^{1/2}\left( {1 - \rho_{i}^{2}} \right)^{1/2}}} & (13)\end{matrix}$

where, W is the signal bandwidth, T is the coherent integration lengthand ρ_(i) is the correlation of the ith baseline. Since the mean errorin AWGN is close to zero, the standard deviation of the error isapproximately the RMS error. The weight is one over the RMS errorsquared, giving a theoretical weighting as

$\begin{matrix}{W_{i} = \frac{1}{\sigma_{i}^{2}}} & (14)\end{matrix}$

These exemplary weighting operations are performed after the field dataanalysis in the “Compute Weights” block 407 as shown in FIG. 4. Othereffects may be included to account for degradations such as multipathwhich further increases the RMS error.

Covariance Matrix Computation

A covariance matrix may be required for making decisions using the jointerror density. The covariance matrix, C, is a port by port matrix of thecovariance between the ith and jth port which is computed as

c _(ij)=β_(ij)σ_(i)σ_(j)  (15)

where, β_(ij) is the correlation coefficient between the ith and jthport from the correlation matrix.

Alternatively this step may be bypassed for computational efficiency ifthe correlation levels between ports are deemed to be too small. Anexemplary decision criterion is to use the covariance matrix if at leastone of the β_(ij) exceeds a correlation threshold. If this threshold isnot exceeded a flag is set to use an independent error analysis.

MAP Decision Metric Computation

The MAP decision computation using the joint error density employs afurther generalization for correlated UTDOA errors. Staring with jointGaussian errors, the a posteriori probability is

$\begin{matrix}{{\ln \left( {f_{L|e}\left( \underset{\_}{L} \middle| \underset{\_}{e} \right)} \right)} = {{\ln \left( {G\; ^{{- \frac{1}{2}}\underset{\_}{e}\; C^{- 1}{\underset{\_}{e}}^{T}}} \right)} = {{\ln (G)} - {\frac{1}{2}\underset{\_}{e}C^{- 1}{\underset{\_}{e}}^{T}}}}} & (16)\end{matrix}$

where, G is a constant. In terms of the individual UTDOA errors

$\begin{matrix}{{\ln \left( {f_{L|e}\left( \underset{\_}{L} \middle| \underset{\_}{e} \right)} \right)} = {{\ln (G)} - {\frac{1}{2}{\sum\limits_{j}{\sum\limits_{i}{e_{i}e_{j}d_{ij}}}}}}} & (17)\end{matrix}$

where, d_(ij) are elements of C⁻¹. Assuming the model in equation (8)for the marginal error probability density, the following generalizationis made to equation (17) for the joint density,

$\begin{matrix}{{{\ln \left( {f_{Le}\left( {\underset{\_}{L}\underset{\_}{e}} \right)} \right)} = {{\ln (G)} - {k{\sum\limits_{j}{\sum\limits_{i}{s_{ij}{{h\left( {r_{i},e_{i}} \right)}}^{p_{i}/2}{{h\left( {r_{j},e_{i}} \right)}}^{p_{j}/2}{d_{ij}}^{{({p_{i} + p_{j}})}/4}}}}}}}\mspace{79mu} {{where},}} & (18) \\{\mspace{79mu} {{h\left( {r_{i},e_{i}} \right)} = \left\{ \begin{matrix}{r_{i}e_{i}} & {e_{i} \geq 0} \\\frac{e_{i}}{r_{i}} & {e_{i} < 0}\end{matrix} \right.}} & (19) \\{\mspace{79mu} {{s_{ij} = {{sgn}\left\{ {h_{i}(r)} \right\} {sgn}\left\{ {h_{j}(r)} \right\} {sgn}\left\{ d_{ij} \right\}}}\mspace{79mu} {and}}} & (20) \\{\mspace{79mu} {{{sgn}(x)} = \left\{ \begin{matrix}1 & {x > 0} \\0 & {x = 0} \\{- 1} & {x < 0}\end{matrix} \right.}} & (21)\end{matrix}$

Substituting (18) and the natural log of (7) into (5) gives

$\begin{matrix}{{\ln \left( {f_{Le}\left( {\underset{\_}{L}\underset{\_}{e}} \right)} \right)} = {{\ln (G)} - {k{\sum\limits_{j}{\sum\limits_{i}{s_{ij}{{h\left( {r_{i},e_{i}} \right)}}^{p_{i}/2}{{h\left( {r_{j},e_{j}} \right)}}^{p_{j}/2}{d_{ij}}^{{({p_{i} + p_{j}})}/4}}}}} + {\ln \left( \lambda_{a} \right)} - {\lambda_{a}D}}} & (22)\end{matrix}$

Since the objective is to find the x,y,z and B that maximizes (22), theterms that do not depend on x,y,z and B can be ignored, giving:

$\begin{matrix}{{\underset{x,y,z,B}{argmax}\left( {\ln \left( {f_{Le}\left( {\underset{\_}{L}\underset{\_}{e}} \right)} \right)} \right)} = {\underset{x,y,z,B}{argmax}{\quad\left( {{{- k}{\sum\limits_{j}{\sum\limits_{i}{s_{ij}{{h\left( {r_{i},{e_{i}\left( {x,y,z} \right)}} \right)}}^{p_{i}/2}{{h\left( {r_{j},{e_{j}\left( {x,y,z} \right)}} \right)}}^{p_{j}/2}{d_{ij}}^{{({p_{i} + p_{j}})}/4}}}}} - {\lambda_{a}{D\left( {x,y,z} \right)}}} \right)}}} & (23)\end{matrix}$

For computational efficiency, equation (23) may be divided by −k and thex,y,z that minimizes

${- \frac{1}{k}}{\ln \left( {f_{Le}\left( {\underset{\_}{L}\underset{\_}{e}} \right)} \right)}$

is found as

$\begin{matrix}\begin{matrix}{{\underset{x,y,z,B}{argmax}\left( {\ln \left( {f_{Le}\left( {\underset{\_}{L}\underset{\_}{e}} \right)} \right)} \right)} = {\underset{x,y,z,B}{argmin}\left( {{- \frac{1}{k}}{\ln \left( {f_{Le}\left( {\underset{\_}{L}\underset{\_}{e}} \right)} \right)}} \right)}} \\{= {\underset{x,y,z,B}{argmin}\begin{pmatrix}\begin{matrix}{\sum\limits_{j}{\sum\limits_{i}{s_{ij}{{h\left( {r_{i},{e_{i}\left( {x,y,z,B} \right)}} \right)}}^{p_{i}/2}}}} \\{{h\left( {r_{j},{e_{j}\left( {x,y,z,B} \right)}} \right)}}^{p_{j}/2}\end{matrix} \\{{d_{ij}}^{{({p_{i} + p_{j}})}/4} + {\frac{\lambda_{a}}{k}{D\left( {x,y,z} \right)}}}\end{pmatrix}}}\end{matrix} & (24)\end{matrix}$

The decision metric to be minimized is then

$\begin{matrix}{M = {{\sum\limits_{j}{\sum\limits_{i}{s_{ij}{{h\left( {r_{i},{e_{i}\left( {x,y,z,B} \right)}} \right)}}^{p_{i}/2}{{h\left( {r_{j},{e_{j}\left( {x,y,z,B} \right)}} \right)}}^{p_{j}/2}{d_{ij}}^{{({p_{i} + p_{j}})}/4}}}} + {\frac{\lambda_{a}}{k}{D\left( {x,y,z} \right)}}}} & (25)\end{matrix}$

For locations where there is low (˜0) cross-correlation betweenbaselines, the covariance matrix is diagonal. Independence is assumedbetween UTDOA errors simplifying (25) to

$\begin{matrix}{M = {{\sum\limits_{i}{\frac{1}{\sigma_{i}^{p}}{{h\left( {r_{i},{e_{i}\left( {x,y,z,B} \right)}} \right)}}^{p_{i}}}} + {\frac{\lambda_{a}}{k}{D\left( {x,y,z} \right)}}}} & (26)\end{matrix}$

In terms of the pre-computed weights for each baseline, the metric is

$\begin{matrix}{M = {{\sum\limits_{i}{W_{i}^{p_{i}/2}{{h\left( {r_{i},{e_{i}\left( {x,y,z,B} \right)}} \right)}}^{p_{i}}}} + {\frac{\lambda_{a}}{k}{D\left( {x,y,z} \right)}}}} & (27)\end{matrix}$

FIG. 14 shows the logic flow for the MAP decision metric computation ateach x,y,z value. The error samples are computed for the x,y,z beingconsidered 1401. In general, an analytical solution for the common bias,“B”, in equation (27) that minimizes M is difficult to compute.Therefore, the biases at two relatively easy to compute sample points(p_(i)=1, r_(i)=1 and p_(i)=2, r_(i)=1) are found and combined toprovide an approximation. A minimum bias is computed assuming Gaussianstatistics 1402 for the errors followed by a computation assuming doubleexponential statistics 1403. The bias to be used is then found bycombining the two bias points 1404. The combining can be done by simplytaking the average of the Gaussian and double exponential biases.Alternatively, the bias for arbitrary p_(i) and r_(i) can be foundthrough a search over all possible biases at the cost of complexity. Ifthis is done, the average of the two bias samples is used as a startingpoint for the search. Alternatively, the search can be done offline andcompared with the bias results for the two bias samples. In this case,the average percentage deviation from the two bias points may be used inthe combining at the cost of offline analysis. After combining thesample biases, the conditional error contribution is determined 1405using the combined bias 1405, the covariance matrix 1407, and the ErrorModel and Parameter table as previously developed from the field data.The computed conditional error contribution 1405 and the previouslydeveloped Model for A Priori Position 1409 are then used to find an apriori contribution 1406 to the MAP metric. The minimum of the metriccomputation in FIG. 14 over x,y,z with various geographical mapresolutions and iterations provides the final solution as depicted inFIG. 4.

Gaussian Bias

The Gaussian bias is found by setting p_(i)=2 and r_(i)=1 in (27). Notethat for r_(i)=1, h(r_(i)=1, e_(i)(x,y,z,B))=e_(i)(x,y,z,B) giving

$\begin{matrix}{M = {\sum\limits_{i = 1}^{N}{\left( {{\Delta\tau}_{i} - B} \right)^{2}W_{i}}}} & (28)\end{matrix}$

where,

N is the number of baselines

Δτ_(i)≡{circumflex over (τ)}_(i)−τ_(i)(x,y,z) is the unbiased error.

A minimum solution over the bias is found by setting the derivative of(28) with respect to B equal to zero and solving for B giving

$\begin{matrix}{B = {\sum\limits_{i = 1}^{N}{{\Delta\tau}_{i}{W_{i}/{\sum\limits_{i = 1}^{N}{W_{i}.}}}}}} & (29)\end{matrix}$

Equation (29) provides a bias when the error distributions are Gaussian.

Exponential Bias

The exponential bias is found by setting p_(i)=1 and r_(i)=1 in (27)giving

$\begin{matrix}{M = {\sum\limits_{i = 1}^{N}{{{{\Delta\tau}_{i} - B}}\sqrt{W_{i}}}}} & (30)\end{matrix}$

Again, a minimum solution over the bias is found by setting thederivative of (28) with respect to B equal to zero and solving for B.The derivative of each term with respect to B is

$\begin{matrix}\begin{matrix}{{\frac{}{B}{{{\Delta\tau}_{i} - B}}\sqrt{W_{i}}} = \left\{ \begin{matrix}\sqrt{W_{i}} & {B \geq \tau_{i}} \\{- \sqrt{W_{i}}} & {B < {\Delta\tau}_{i}}\end{matrix} \right.} \\{= {\sqrt{W_{i}}\left\lbrack {{U\left( {B - {\Delta\tau}_{i}} \right)} - {U\left( {{\Delta\tau}_{i} - B} \right)}} \right\rbrack}}\end{matrix} & (31)\end{matrix}$

where, U(x) is a unit step function (see A. Oppenhem and A. Willsky,Signals and Systems, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1983).Setting the derivative of the sum equal to zero gives

$\begin{matrix}{\frac{M}{B} = {{\sum\limits_{i = 1}^{N}{\sqrt{W_{i}}\left\lbrack {{U\left( {B - {\Delta\tau}_{i}} \right)} - {U\left( {{\Delta\tau}_{i} - B} \right)}} \right\rbrack}} = 0}} & (32)\end{matrix}$

Each term in (32) as a function of B is −√{square root over (W_(i))}until the value Δτ_(i) is reached and then there is a step to −√{squareroot over (W_(i))} for B greater than Δτ_(i). Due to thesediscontinuities, there is no exact solution for B. However, a value forB can be found that provides an approximate solution.

A solution in FIG. 15 orders the summation in (32) according toincreasing Δτ_(i). The value of B is then found that makes (32) as closeas possible to zero. This occurs at a value of B where the kth steptransition has occurred making the sum of the negative termsapproximately equal to the sum of the positive terms (i.e.

${\sum\limits_{i = 1}^{K}\sqrt{W_{i}}} \approx {- {\sum\limits_{i = {K + 1}}^{N}\sqrt{W_{i}}}}$

where N is the total number of baselines in the ordered summation). Inthe figure the weight and sample arrays are populated and sorted. Athreshold is computed that is

$\sum\limits_{i = 1}^{N}{\sqrt{W_{i}}/2}$

to provide a stopping condition. The terms are accumulated in order fromthe terms with the smallest to the largest transition point. At thepoint where the threshold is reached, the value Δτ_(K) is returned ifthere are an odd number of terms; otherwise, the kth term's transitionpoint is averaged with the prior term's transition point.

Metric Calculation

The first term in (27) is computed following the steps in FIG. 16. Foreach baseline, the measurement parameters such as UTDOA correlation, thenumber of baselines and multipath parameters are determined. Thesemeasurement parameters are used to determine error model parameters,p_(i) and r_(i), from a lookup table or through a direct calculationusing a parameter fitting model. The errors are adjusted depending onthe skew. Finally, the summation in the metric is computed using (25)when the errors have significant correlation or using (27) when theerrors do not have significant correlation. The significance of thecorrelation is determined as part of the covariance matrix computationsin FIG. 4. Finally, the last term in (27) and (25) is computed using (6)to account for the a priori probability.

Sample Results

Table 1 below illustrates sample improvements relative to a weightedleast squares algorithm. Using approximately 46,000 locationmeasurements, the distribution of positioning errors were compiled usingthe weighted least squares algorithm and the algorithm above. Theparameters for the above model were chosen using a separate trainingdata set of 32,000 locations. The table shows improvement ofapproximately 20 meters and 2 meters in the 95th and 67^(th) percentilesrespectively. The average error improved by approximately 15 meters.

TABLE 1 Exemplary Results Relative to Weighted Least Squares MethodError (m) Weighted Least Squares MAP 67^(th) percentile 74.7 72.995^(th) percentile 293.1 271.3 Average 287.9 272.6

CONCLUSION

The present invention, and the scope of protection of the followingclaims, is by no means limited to the details described hereinabove.Those of ordinary skill in the field of wireless location willappreciate that various modifications may be made to the illustrativeembodiments without departing from the inventive concepts disclosedherein.

1. A method for use in a wireless location system, comprising: obtainingfield data, wherein said field data have baseline or location dependentvalues to be used in a signal correlation model; analyzing said fielddata to obtain (1) said signal correlation model and associatedmeasurement parameters, (2) correlation matrix rules, and (3) a modelfor a priori position; computing weights for the measurements based onan estimated variability of the measurement; using the weights alongwith the correlation matrix rules to generate a covariance matrix, andcomputing an inverse covariance matrix; performing an iterative searchover a geographical region to find a location with a maximum aposteriori (MAP) metric; determining that a stopping condition has beenreached; and reporting the geographic position with the largest MAPmetric; wherein said iterative search includes a resolution loop inwhich a geographic search space resolution is reduced in each iterationand new test points are generated via interpolation, and wherein saiditerative search includes a MAP metric computation that uses thecovariance matrix, an error model and a measurement parameter table. 2.A method as recited in claim 1, further comprising generating a tableproviding a mapping between the measurement parameters for a skew and acorrelation value for the measurement.
 3. A method as recited in claim1, further comprising generating a table providing a mapping between themeasurement parameters for a skew and the number of baselines.
 4. Amethod as recited in claim 1, further comprising analyzing thecorrelation between different receiver ports linking the locationreceiver to an external antenna, and providing correlation values andrules for their application.
 5. A method as recited in claim 1, whereinthe search is re-centered at the previous iteration's minimum errorpoint before proceeding, wherein test points within the currentgeographical region search space are individually searched and a MAPmetric is computed for each test point.
 6. A method as recited in claim1, wherein the method is performed in a wireless location system (WLS)employing at least one of: a time of arrival (TOA) location algorithm,an uplink time difference of arrival (TDOA) location algorithm, adownlink TDOA algorithm, an angle of arrival (AOA) location algorithm,and a hybrid location algorithm.
 7. A method as recited in claim 6,wherein the downlink time difference of arrival location algorithmcomprises the use of downlink satellite beacons from a globalpositioning system (GPS).
 8. A method as recited in claim 7, wherein themethod is performed in a WLS employing an assisted GPS location system.9. A method as recited in claim 6, wherein the hybrid location algorithmemploys uplink signals from a mobile device and downlink signalsreceived at the mobile device.
 10. A method as recited in claim 1,wherein the error model comprises an error distribution and the fielddata is analyzed to obtain the error distribution.
 11. A method asrecited in claim 10, wherein the measurement parameters are analyzed toidentify which measurement parameters cause large changes in the errordistribution and determining ranges and bin sizes for the identifiedmeasurement parameters.
 12. A method as recited in claim 11, furthercomprising: applying a coarse modeling step comprising computing valuesfor error model parameters that match the field data; and applyingresults of the coarse modeling step to update the error model.
 13. Amethod as recited in claim 12, further comprising applying a finemodeling step and applying results of the fine modeling step to updatethe error model.
 14. A method as recited in claim 13, wherein said finemodeling step comprises accumulating conditional statistics over thefield data based on ranges and bin sizes.
 15. A method as recited inclaim 14, further comprising determining a conditional error bycomputing a skew ratio based on the mean and standard deviation of theerror for each measurement parameter bin and selecting as theconditional error a center of the bin.
 16. A method as recited in claim1, wherein said performing an iterative search further comprises:approximating a common bias by computing a first bias point by assumingGaussian statistics for the errors and a second bias point by acomputation assuming double exponential statistics; and determining thecommon bias by combining the first and second bias points.
 17. A methodas recited in claim 16, further comprising: determining a conditionalerror contribution using the common bias, the covariance matrix, theerror model, and the measurement parameter table.
 18. A method asrecited in claim 17, wherein said second bias point is computed by:populating and sorting weight and sample arrays; computing a thresholdas $\sum\limits_{i = 1}^{N}{\sqrt{W_{i}}/2}$ to provide the stoppingcondition; accumulating the terms in order from terms with the smallestto the largest transition point; and at the point where the threshold isreached, returning Δτ_(K) when there are an odd number of terms andotherwise averaging the kth term's transition point with the priorterm's transition point.
 19. A method as recited in claim 3, whereinsaid performing an iterative search further comprises: determiningmeasurement parameters for each baseline and using the measurementparameters to determine error model parameters; and updating the errormodel parameters as a function of the skew.
 20. A method as recited inclaim 19, wherein the error model parameters are determined from alookup table or using a parameter fitting model, wherein a summation ofthe MAP metric is computed using the covariance matrix when the errorshave significant correlation, and wherein the summation of the MAPmetric is computed using a fast matrix calculation when the errors donot have significant correlation.
 21. A system, comprising: means forobtaining field data, wherein said field data have baseline or locationdependent values to be used in a signal correlation model; means foranalyzing said field data to obtain (1) said signal correlation modeland associated measurement parameters, (2) correlation matrix rules, and(3) a model for a priori position; means for computing weights for themeasurements based on an estimated variability of the measurement; meansfor using the weights along with the correlation matrix rules togenerate a covariance matrix, and computing an inverse covariancematrix; means for performing an iterative search over a geographicalregion to find a location with a maximum a posteriori (MAP) metric;means for determining that a stopping condition has been reached; andmeans for reporting the geographic position with the largest MAP metric.